Proceedings of the American Mathematical Society, 131 (2003) 3, 825-834, also a talk at the Joint Mathematics Meeting in January 2001.
Lomonosov's invariant subspace theorem states that a linear operator T that commutes with a compact operator C (i.e., TC=CT) has a nontrivial invariant subspace L (i.e., T(L)Í L). The proof makes use of the Schauder fixed point theorem. The idea of the present paper is to prove a similar theorem for multivalued linear operators by means of a fixed point theorem for multivalued maps, e.g., the
Kakutani Theorem. Then a (single valued or multivalued) linear operator has an invariant subspace if it commutes with a multivalued "compact" operator.
Preprint in pdf (10 pages).